Abductive Reasoning by Children

Review of Psychology Frontier

(RPF)

Abductive Reasoning by Children

Heidi Kloos 1 , Guy Van Orden 2

CAP Center for Cognition, Action, and Perception, Department of Psychology, University of Cincinnati

Cincinnati OH, USA

'heidi.kloos@uc.edu; 2 guy. van. orden® uc.edu

Abstract- Children can link facts and events into integrated
beliefs. This ability of the mind to combine facts to form higher-
order Gestalts is central to many cognitive activities, including
problem solving, analogical reasoning, and creative thought. In
fact, it is central to the abduction of meaning: the creation of a
self-sustaining pattern of ordered facts that are combined in the
larger Gestalt. Abduction has mostly escaped experimental
investigation, possibly because it often emerges instantly and
non-linearly, and is thus difficult to trace with traditional models
of cognition. In the current paper, we take steps towards filling
this gap, using ideas from nonlinear dynamics and complexity
science. The assumption is that products of abductive reasoning
can emerge from competing sources of constraint, namely
constraints that favor local facts (and contradict a congruent
Gestalt) versus constraints that favor the congruent Gestalt (and
override contradictory local facts). The experiments reviewed in
this paper exploit situations of such conflicting constraints. The
goal is, first, to provide evidence of congruent-Gestalt constraints
in young children, and second, to explore the interaction among
competing constraints. The outcome is a qualitative evaluation of
parameter dynamics, the dynamics of a control parameter of
abductive reasoning.

Keywords-Abduction; Reasoning; Nonlinear Dynamics;
Constraints; Parameter Dynamics

I. INTRODUCTION

Charles Saunders Pierce coined the term abduction to refer
to the essential capacity of a person to form innovative
interpretations, to bring together otherwise separate empirical
facts or events (cf. [2] ). Abduction has been linked to mental
phenomena such as insight and the discovery of facts as
patterns in data, abstraction of hidden properties, diagnosis of
causes of events, and the evaluation of competing
explanations (e.g., for reviews see [ ' 34] ). Yet, the empirical
study of abduction has focused largely on adult reasoning,
with little explicit investigation of abductive reasoning in
children. This is surprising given that children early on can
organize facts into coherent ideas (e.g., [9 ' "' 15 ' 22 ' 40, 41] ). For
example, they can make causal inferences after only short
demonstrations (e.g., see , for a review), and they can form
beliefs about the behavior of objects in laboratory
demonstrations [14] . Even infants appear to abduct ideas about
systematic patterns over time, ignoring features that do not fit
within those patterns (cf. [1 ' 7] ).

Perhaps these early attempts of children to create meaning
are not be sufficiently rational to fit the common definition of
abduction. They might be based on associative processes with
little explicit hypothesis generating on the part of a child (cf.,
[29] ). We nevertheless should not rule them out as abductive
reasoning, given that they lead to unified beliefs and causal
explanations. In other words, it might not be necessary to tie
abductive reasoning to explicit rationality. Following Pierce's
definition, we define abduction instead in terms of its product,
not the hypothesized cognitive process that gives rise to
abduction. Specifically, abduction is the emergence of a
coherent organization among facts and events. Such

emergence of coherence could happen instantaneously,
analogous to a Gestalt phenomenon [351 , or it could involve the
piece-meal construction of relations one by one (cf. [5 ' 6I ). And
emergent coherence can involve different levels of abstraction,
ranging from what we commonly think of perception to what
is typically discussed under abstract thought, all the way to
explicit comparisons of hypotheses. This necessarily broad
view of abduction makes it possible to develop a description
of abductive performance without first assuming a particular
cognitive process. The hope is to better understand how
young children organize separate facts and events into the
larger wholes of coherent ideas.

To describe abductive reasoning, we borrow ideas from
the framework of nonlinear dynamics and complexity science.
This framework has been applied repeatedly to questions of
children's development, including motor development (e.g.,

r-> f-t nil

), the A-not-B error (e.g., ), early language development
(e.g., [37] ), spatial reasoning (e.g., [2?I ), and problem solving .
The general idea is that a qualitative change in performance
need not be singly caused, but instead results from changes
among multiple sources of constraints favoring one or another
performance outcome. We first illustrate what we mean by
constraints and then review empirical findings with children
to demonstrate how a constraint-based description can capture
a large body of findings.

A. Constraints and Control Parameters

Our starting point is the assumption that multiple sources
of constraint determine the degrees of freedom for thought
and action, constraints that can be summarized in control
parameters. Applied to infant stepping behavior, for example,
the sources of constraints pertain to gravity and the mass of an
infant's legs on the one hand, and the muscle strength of the
legs and the infant's willful control of the legs on the other
hand. These constraints can be summarized in a ratio of
competing forces to define an idealized control parameter of
stepping behavior (cf., [20 ' 391 ). In other words, a control
parameter for stepping behavior is concisely summarized by
the ratio of the weight of the baby's legs relative to the
strength of the baby's legs. Despite being a simplification of
the larger system in which control is realized, the data fit this
conceptualization: Soon after being born, when the legs are
still light, most infants can raise their legs in stepping
behaviour (i.e., [lighter legs]/ [strong enough muscles] =
[stepping behavior]). However, as the infant grows, the
concomitant increases in leg mass may exceed the increase in
the legs' muscle strength, eliminating stepping behaviour for a
period of time (i.e,. [heavier legs]/ [not strong enough muscles]
= [no stepping behavior]). Later, the increasingly stronger leg
muscles reach a point wherein the strength of the legs
surpasses their mass, allowing the child to step (i.e., [heavier
legs]/ [strong enough muscles] = [stepping behavior]).

To understand abduction, we seek a similar ratio of
constraints that we envision as a control parameter capturing

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the tradeoff among constraints that might bring about
qualitative changes in a child's thought. As mentioned,
abduction brings about congruent Gestalt-like organizations
of facts and events. We take this definition to imply the
existence of constraints that favor such global or macro-scales
of organization and the first experiment corroborates that
these constraints exist in young children. We then exploit
circumstances in which constraints favoring congruent
Gestalts indeed lead to congruent thought, although they do
not exist in the facts at hand. We explore manipulations
affecting the salience of local facts, which turn affect
children's creation of false Gestalts (i.e., Gestalts that
compete with local facts).

Like stepping behavior, then, abduction may include a
competition among mutually exclusive constraints —
constraints on different levels of organization that cannot both
met at the same time. The following formula summarizes the
tradeoff between such competing constraints as a ratio of a
control parameter A:

Constraints Sustaining Integrity of Local Facts
A = (1)

Constraints toward Global Congruence among Facts

To make these ideas more concrete, consider a scenario in
which the local elementary fact pertains to a person's attitude
towards traveling. The person might like to travel, or not -
neither attitude by itself inviting a higher-order Gestalt. Now
add a second person to the scenario. This second person's
attitude towards traveling might match with that of the first
person (e.g., both individuals like travelling), or not (e.g., one
person likes to travel, while the other person does not).
Congruent attitudes arise in the first case, when both people
feel the same towards traveling, but not in the second case,
when only one of them likes to travel.

To take this scenario one step further, add the attitude of
one person towards the other. The two individuals might like
each other, or dislike each other. Each of the individual
attitudes (i.e., towards each other and towards traveling) can
be thought of as local facts. But when combined, an even
higher-order congruency becomes possible, namely that of
transitive congruence among three attitudes. This congruent
Gestalt is present, for example, when the two people like each
other and they both like traveling. It is also present when the
two people like each other, and they both dislike traveling, or
when the two people dislike each other, and they disagree
about traveling. Figure 1 shows two congruent attitude
examples in schematic form, as well as two incongruent
examples, namely when the two people like each other, but
have opposite attitudes towards traveling.

The scenario of attitudes illustrates several aspects of what
is called a control hierarchy (cf. [381 ). First, sources of
constraint are nested hierarchically, from the most elementary
to the most global. Individual attitudes are treated as local
elements, nested within the pattern of two-relation match,
nested in turn within the pattern of three -relation congruence.
One can even imagine that an isolated attitude is a pattern of
some sort, namely one that combines individual instances of
travel events into a unified belief. And one can even image
that the three -relation congruence is an element of some sort,
for example of a higher-order theory of relationships stability.
In studying abduction, one may demarcate a particular level
of unified facts, with the idea that the same principles should
hold when another level is considered in the hierarchy of
nested Gestalts.

Second, the scenario of attitudes illustrates how
constraints may affect abduction. As long as only one single
attitude is considered, constraints toward reproducing that
single fact are straightforward, deriving from what we know
about remembering single facts. But when two or more facts
are to be kept in mind, abduction of higher-level relationships
becomes possible, and interactions between facts may shape
the abductive outcomes. For example, it is easier to remember
congruent facts than facts that create some mismatch among
each other (cf, [101 ). In fact, constraints toward forming
congruent relations among facts may even overwhelm
constraints that otherwise sustain elementary facts of the
matter, in which case elementary facts are misremembered in
favor of a congruent abductive outcome. In this latter scenario,
the constraints favoring veridical elementary facts oppose the
constraints favoring congruent relations among facts, creating
the ratio of opposing constraints illustrated in Formula 1.

Mary — likes Victor Mary

Victor

Traveling

Mary — likes

Victor Mary

\

\ /
Not Traveling

likes Victor

Traveling

Not Traveling

Traveling
Not Traveling

Fig. 1 Schematic representation of attitudes that are either congruent among
each other (marked with a red circle) or incongruent (marked with a blue x-ed

circle)

'Mary' and 'Victor' stand for two persons who either like or dislike each
other, and who each either like or dislike traveling.

B. Overview of Reviewed Studies

In what follows, we describe five studies that examined
children's capacity to remember correctly the details of local
facts'. First, by reducing or eliminating constraints that favor
the integrity of local facts, we shed light on the constraints
that favor abduction of congruent higher-order Gestalts.
Second, by pitting weak versus strong local constraints
against the constraints that favor congruent Gestalts, we
reveal the flexible nature of the child's mind to either abduct
the higher-order congruent Gestalt or not.

Instead of attitudes, the reviewed studies use feature
correlations as elementary facts. Replace the previous social
world of two people and their feelings about travel, with a
world of objects that differ in mass, volume, and the rate at
which they sink to the bottom of a water tank. A set of
plausible local facts about these objects might include a
positive correlation between mass and volume (e.g., heavier is
bigger), a positive correlation between mass and sinking
speed (e.g., heavier is faster), and a positive correlation
between volume and sinking speed (e.g., bigger is faster).
These three local facts are globally congruent among each
other, as shown schematically in Figure 2a. Other examples of

Parts of these findings were published in

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Review of Psychology Frontier

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congruent facts include heavier is bigger, heavier is slower,
and bigger is slower (Figure 2b); or heavier is smaller,
heavier is faster, and smaller is faster. But a single change in
the facts in evidence can eliminate the globally congruent
order. For example, a set of locally possible but globally
incongruent facts would be heavier is bigger, heavier is faster,
and bigger is slower (see Figure 2c).

heav

slow

Fig. 2 Schematic representation of feature correlations, each pertaining to a
set of objects

The correlations are either congruent among each other (marked with a red
circle) or incongruent (marked with a blue x-ed circle).

A general outline of our predictions was derived using
Formula 1. The presence of constraints sufficiently strong to
sustain local facts yield a numerator greater than its
denominator, therefore yielding a value f or A > 1 . In contrast,
the presence of weak local constraints, insufficiently strong to
sustain local facts, would yield a numerator smaller than the
denominator, yielding a value for A < 1. The A = 1 case
reflects either a case in which local and global constraints are
aligned together to support veridical performance equivalently.
Or it reflects a case in local and global constraints oppose
each other, but are exactly equal in strength. Both of these
cases are uninteresting for the current purposes. The former
one would make it impossible to determine whether children
have detected the Global patterns. And the latter case is an
unstable, so-called saddle point that degenerates into A > 1 or
A < 1 with the slightest perturbation. It would produce a
random pattern of performance respecting neither local nor
global constraints overall.

Local constraints can be brought under experimental
control in order to test for empirical outcomes consistent with
our predictions derived from Formula 1. We review five
experiments that implement such manipulations. The first
experiment creates a scenario in which local constraints are
missing altogether (i.e., children have to make a guess a about
a local fact). In the second experiment, local constraints are
weak, in that children have to learn two opposite correlations
(analogous to learning that two people have opposite attitudes
towards travel). The third experiment adds a condition in
which local constraints are strong, in that children have to
learn two matching correlations (analogous to learning that
two people feel the same about travel). Finally, the last two
experiments replicate the no-local-constraints and the weak-
local-constraints scenarios, respectively, using a new set of
facts - ones that are improbable, as a means of minimizing the
influence of participants' preconceptions about what they are
asked to learn. For each experiment, we first review the
method and then describe the findings.

II. Review of Experiments

A. Experiment 1: No Local Constraints

The method includes three main steps, two of which are
designed to teach 4 to 5 -year-old about feature correlations,
and the third one designed to test their beliefs at the end of the
training phase. In particular, in the first step, a preschooler is
shown a set of 'submarine' cylinders that differ in mass and
volume in such a way that the heavier submarine is also the
larger one. The child then engages in activities to learn the
fact that heavier is bigger. In the second step, the child
watches two submarines race to the bottom of a water tank,
taking note of the submarine that arrives at the bottom to the
tank first. The two cylinders differ only in mass, and they
convey the fact that heavier is faster. In the final step, the
preschooler is invited to play against a submarine man, the
fantasy creature who built the submarines. The game is to
design a submarine that will sink faster than the submarine of
the submarine man - for example, by the child adding more
weight to the submarine. Figure 3a gives a schematic
representation of what such a mass trial looks like: the
submarine of the submarine man is the standard, and the child
can choose either a heavier weight (square with more lines) or
a lighter weight (square with fewer lines), while the volume of
the child's submarine is the same as that of the standard.

Standard

Test Options

B. Volume Trial

aD

B
Test Options

Fig. 3 Schematic representation of the test items presented to children in
the third step of Experiments 1 and 2

To create a ' submarine' that would sink faster than the standard, children can

either adjust the mass of their submarine (choosing between a heavier and a

lighter option; A), or they can adjust the volume of their submarine (choosing

between a larger and a smaller options, B).

On the basis of the chosen options across trials, we can
ascertain the child's belief of the sinking fact. For example, if
the child consistently chooses the heavier option to make
faster submarines, the child exhibits the fact that heavier is
faster. Importantly, the child is also presented with volume
trials during this third step (Figure 3b), trials in which choices
pertain to the volume of the submarine, not mass. In other
words, while the child has observed the effect of only one
feature (e.g., mass) on sinking speed, testing includes both
mass trials and volume trials. And mass trials are intermixed
with volume trials, with no explicit instructions about having
to make a guess. The child is simply asked to build a
submarine that will win against the submarine man's.
Nonetheless the absence of demonstrations about the local
volume-sinking fact opens the door to a value of A < 1, which
necessitates the constraints of abductive reasoning toward
global congruence.

To establish the existence of these global constraints,
children participated in one of four conditions that differed in
the mass-volume fact and the demonstrated sinking fact (for a
schematic of the conditions, see Figure 4). The demonstrated
sinking facts are shown as solid lines in Figure 4. And the to-
be-guessed sinking facts are shown as dashed lines with two
question marks. The crucial test was whether children's guess
about the unknown sinking fact is constrained by the

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possibility of a global congruence among all three feature
correlations. If constrained by congruence, the children who
learned that heavier is bigger and heavier is faster should
guess that bigger is faster too. And the children who learned
that heavier is bigger and bigger is slower should guess that
heavier is slower too. Conversely, the children who learned
that heavier is smaller and heavier is faster should guess that
smaller is faster. And the children who learned that heavier is
smaller and smaller is faster should guess that heavier is
faster.

Mass-Volume Fact

heavy = big

heavy = small

Sinking Fact:
heavy = fast

Sinking Fact:

big = slow

(small = fast)

Fig. 4 Schematic representation of the four conditions of Experiment 1

The guessed fact in each condition is marked with question marks. The other
two facts are presented to children.

In line with the predictions of weak local constraints,
preschoolers were highly likely to abduct a globally congruent
set of facts. Figure 5 shows scatterplots of children's choices
for the sinking facts, separated by condition. The individual
points of each scatterplot correspond to individual children (N
= 8 per condition) and portray the proportion of "heavier"
choices on mass trials (X-axis) and the proportion of "bigger"
choices on the volume trials (Y-axis). Red circles mark the
corners of the scatterplot where a child's choices would be
100% globally congruent. As can be seen in the figure, almost
all of the children's performance fell within the red circles (30
out of 32). In other words, almost all of the children made
systematic choices consistent with globally congruent facts
anticipated from A < 1 .

Mass-Volume Fact

heavy = big heavy = small
® Q —

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0.25 0.5 0.75 1

-0

D

CD

3
o

— K-

0.25 0.5 0.75 1

Proportion of 'More' Choices

® © «™

0-

Proportion of 'More' Choices

-©

D

CD

3
o

Mass Trials:
Proportion of 'Heavier'Choices

Fig. 5 Scatterplots of individual children's responses in Experiment 1
(N = 8 per cell), measured in proportion of 'heavier' choices on mass trials
and proportion of 'bigger' choices on volume trials, separated by condition

The red circles denote the corners of the scatterplot that correspond to
congruent facts.

In sum, guessed sinking facts were almost always
congruent with the other two facts, bringing mass-volume,
mass-speed and volume-speed into Gestalt-like global
congruence. Given that children were open to guess one of the
facts, the local constraints of that fact were weak. Yet,
children did not produce a random dispersion of choice
responses; they did not guess blindly. Instead their choices
were systematic consistent with constraints favoring
abductions toward global congruence. This finding establishes
the very existence of this higher-order constraint in preschool
children. The next two experiments expand on this result by
manipulating the relative strength of local constraints.

B. Experiment 2: Weak Local Constraints

Children between 4 and 7 years of age participated in a
three-phase procedure that closely mimicked Experiment 1.
They were first shown a set of submarines that differ in mass
and volume (to convey a mass-volume fact). And they were
then shown a series of submarines, racing in pairs to the
bottom of the water tank (to convey sinking facts). Finally,
their inductions were assessed in mass trials and volume trials,
presented in random order. The difference from Experiment 1
was that sinking demonstrations conveyed facts about mass
(while volume was held constant) and volume (while mass
was held constant). That is to say, children saw pairs of
submarines in which the heavier one sank fastest (i.e., heavier
is faster); and they saw pairs of submarines in which the
bigger one sank more slowly (or the smaller one sank faster;
smaller is faster).

Note that the two demonstrated sinking facts are feature
correlations with opposite signs. Mass and sinking speed
follow a more-is-more (positive) relation, while volume and
sinking speed follow a less-is-more (negative) relation. Given
the heightened cognitive demand for learning opposite facts,
we predicted that this scenario could yield weaker local
constraints, and thus we should see children's choices during
mass trials and volume trials to be constrained by global
congruence.

Incongruent Facts Congruent Facts

heavy

big

small heay

small

fast

Fig. 6 Schematic representation of the two conditions of Experiment 2
(incongruent-facts vs. congruent-facts condition)

The conditions differ in whether the same two sinking facts (heavier is faster
and smaller is faster) are congruent with the mass-volume fact (congruent-
facts condition) or not (incongruent-facts condition). Given that children are
asked to remember two sinking facts of opposite relations (one being positive
and one being negative), they are expected to be constrained by higher-order
congruence.

To test this prediction explicitly, children participated in
one of two conditions. The conditions were identical in the
demonstrated sinking facts, but differed in whether the two
sinking facts were congruent with the mass-volume fact
presented to children in the first phase. Figure 6 shows a
schematic of the two conditions. Specifically, in the globally
incongruent-facts condition, the facts were incongruent (i.e.,

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heavier is bigger, heavier is faster and smaller is faster). And
in the globally congruent-facts condition, the facts were
congruent (heavier is smaller, heavier is faster, and smaller is
faster). If children attend to the higher-order pattern of global
congruence, they should make systematic mistakes in the
incongruent-facts condition (because local facts conflict here
with higher-order congruence, yielding A < 1). And they
should be able to learn the (same) local facts in the congruent-
facts condition (because here local facts are aligned with
higher-order congruence, yielding A = 1).

Incongruent Facts Congruent Facts

-3) Q-

0.25 0.5 0.75

Mass Trials:

Proportion of Heavier' Choices

Fig. 7 Scatterplots of individual children' s responses in Experiment 2
(N = 8 per cell), measured in proportion of 'heavier' choices on mass trials
and proportion of 'bigger' choices on volume trials, separated by condition

The red circles denote the corners of the scatterplot that correspond to

congruent facts. And the grey arrows denote the corners of the scatterplot that

correspond to correct facts.

Figure 7 shows the scatterplots of children's individual
performance, separated by condition (incongruent-facts,
congruent-facts) and age group (4-5 vs. 6-7) (N = 8 per cell).
Like before, the scatterplots portray the proportion of
"heavier" choices on mass trials (X-axis) and the proportion
of "bigger" choices on the volume trials (Y-axis). Red circles
mark the corners of the scatterplot where choices would be
100% globally congruent. And gray arrows mark the corners
of the scatterplot where choices would be 100% correct,
reflecting the demonstrated sinking facts. As can be seen in
the figure, most children produced choices that were
congruent. This means that a majority of children in the
congruent-facts condition produced choices that were
factually correct (62% of 4-5 year-olds and 75% of the 6-7
year-olds), while none of the children did so in the
incongruent-facts condition. Most children in this latter
condition were misled to create globally congruent facts
instead (69% of 4-5 year-olds and 88% of 6-7 year-olds).

In sum, findings from this second experiment agree with
the findings from Experiment 1 . Under weak local constraints,
children made the predicted systematic choices to establish
congruent facts in the incongruent fact condition.

C. Experiment 3: Strong Local Constraints

So far we have shown that young children are biased
toward congruent facts when local constraints were either

absent (Experiment 1) or weakened (Experiment 2). In the
next experiment we add a condition in which constraints
towards local facts were strong. The general procedure
mimics the three-phase procedure of Experiment 2:
Participants were first shown a set of submarines that differ in
mass and volume (to convey a mass-volume fact). And they
were then shown a series of submarines racing in pairs to the
bottom of the water tank (to convey sinking facts). Finally,
their abductions were assessed in mass trials and volume trials,
presented in random order in the final phase.

Different from Experiment 2, the details of the facts about
sinking relations were manipulated, as a means of
manipulating the strength of local constraints. In particular,
the two sinking facts had opposite direction in the weak-local-
constraints condition (heavier is faster and small is faster;
identical to Experiment 2), and the two sinking facts matched
in direction in the strong-local-constraints condition (heavier
is slower and bigger is slower). Figure 8 presents a schematic
of these two conditions.

Weak Local
Constraints

Strong Local
Constraint

heavy

big
small

heavy

fast

slow

Fig. 8 Schematic representation of the two conditions of Experiment 3 (weak-
local -constraints vs. strong-local-constraints condition)

Both conditions involved incongruent facts. The difference pertained to

whether participants had to learn two sinking facts of opposite directions (left

graph) or of matching directions (right graph).

Note that the two sinking facts were incongruent with the
mass-volume fact in both conditions. Thus, local constraints
were pitted against global constraint. Thus, learning two
sinking facts of matching directions increase the ratio of
constraints such that A > 1 . And learning two sinking facts of
opposite relations decrease the ratio of constraints to A < 1.
We should therefore expect to see systematically mistaken
choices to create global congruence in the weak-local-
constraints condition, but less so in the strong-local-
constraints condition. Participants included children between
5 and 9 years of age, as well as adults. Figure 9 shows the
scatterplots of individual participants' performance (again
plotting mass-trials performance against volume-trials
performance), separated by condition (weak- vs. strong-local-
constraints) and age group (5-9-year-olds vs. adults).

Red circles mark the corners of the scatterplot where
choices would be 100% globally congruent; and gray arrows
mark the corners of the scatterplot where choices would be
100% correct with the demonstrated sinking facts. Note first
the patterns of responses of the children (top row), focusing
on the correct versus congruent corners of the graph (gray
arrows vs. red circles). Confirming our prediction, children in
the weak-local-constraints condition were more likely to
create congruent facts than children in the strong-local-
constraints condition. And vice versa, children in the weak-
local-constraints condition were less likely to produce correct
facts than children in the strong-local constraints condition.

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Weak Local Strong Local

Constraints _ ^ Constraints

*e i e-

0.25 0.5 0.75 1 . 25 0.5 0.75 1

Mass Trials:
Proportion of 'Heavier' Choices i

Fig. 9 Scatterplots of individual participants' responses in Experiment 3,

measured in proportion of 'heavier' choices on mass trials

and proportion of 'bigger' choices on volume trials, separated by condition

and age group

The red circles denote the corners of the scatterplot that correspond to

congruent facts. And the grey arrows denote the corners of the scatterplot that

correspond to correct facts.

Adults, by contrast, were not affected by constraints
toward congruence in either of the conditions (bottom row of
Figure 9). Their choices correctly reproduced the
demonstrated sinking facts whether these facts were sustained
by weak or strong constraints, suggesting that their
susceptibility to creating global congruent facts differs from
that of children. Decreased susceptibility to higher-order
congruence may come in part from more stable beliefs about
physics (e.g., the belief that heavier objects cannot possibly
sink more slowly than lighter objects). Watching
demonstrations in which this belief was contradicted (strong-
local-constraints condition) might have weakened local
constraints.

To sum up, the goal of Experiment 3 was to test for
changes in performance due to the strength of local
constraints, and we confirmed the expected interplay between
local and global constraints. Experiment 3 also included a
direct replication of findings from Experiment 2 across a
wider age range: 5 to 9-year-old made choices creating
higher-order congruence, in the face of weak local constraints.
Lastly, adults differed from children, making choices
consistent with demonstrated facts in all cases. The final two
experiments are conceptual replications using arbitrary
relations as facts.

D. Experiment 4: Conceptual Replication I

"A transformer was found on a far-away planet. If you put
something in on one end, something different will come out
on the other end." This cover story justified local facts that
combine completely arbitrary features. The specific features,
chosen to have little or nothing in common, pertained to (1)
the size of a cartoon mouse, (2) the darkness of a cloud, and
(3) the depth of a bowl (with no change in volume). Note that
these features have dimensional properties, in that they have a

'more' pole (big; dark; deep), and a 'less' pole (small; light;
shallow) 2 . As such we could create the fact of a 'positive'
correlation (e.g., bigger is darker) or the fact of a 'negative'
correlation (e.g., bigger is lighter). These facts were
demonstrated through a series of movies, showing, for
example, a big or a small mouse entering the transformer and
emerging as a dark or a light cloud, respectively.

The design was a combination of Experiment 1 (when
participants had to guess a fact) and Experiment 3 (when they
were presented with two facts that either matched in direction
(strong-local-constraints condition) or had opposite directions
(weak-local-constraints condition). The general procedure
consisted of a demonstration phase and a testing phase.
During demonstrations, participants were presented with
movies conveying two separate facts. For example,
participants learned a size-darkness fact and a depth-darkness
fact. Importantly, in the strong-local-constraints condition,
these two facts matched in direction (e.g., bigger is lighter,
deeper is lighter ). And in the weak-local-constraints
condition, these two facts had opposite directions (e.g., bigger
is darker, deeper is lighter). Analogous to Experiment 3, the
idea was that opposite facts yield weaker local constraints
than matching facts. As a result, the matching-facts scenario
yields a ratio of constraints A > 1, while the opposite-facts
scenario yields a ratio of A < 1 . We therefore expected to see
more choices creating congruent facts in the weak-local-
constraints condition (when children are exposed to opposite
facts) than in the strong-local-constraints condition (when
children are exposed to matching facts).

During testing, participants were presented with choices to
decide on a particular fact. For example, they were presented
with a dark-grey cloud and a light-grey cloud and asked:
"Which cloud will the big mouse turn into?" Importantly,
while only two facts were demonstrated to participants,
beliefs about all three facts were assessed, requiring that one
fact be guessed. Using the example from above, participants
who saw the size-darkness fact and the depth-darkness fact in
demonstrations were not only asked about these two facts
during testing, but also asked about the size-depth fact, which
was not demonstrated beforehand.

We again created scatterplots displaying the proportion of
'more' ('darker,' 'deeper,' or 'bigger') choices made by a
participant (see Figure 10). Red circles represent choices
yielding 100% congruent facts 4 . As can be seen in the figure,
children were rather overwhelmed by having to learn arbitrary
relations: many of them did not make consistent choices

Note that the identified 'more' pole is somewhat arbitrary in the
case of darkness and depth. For achromatic color, for example, one
could easily imagine the 'more' pole to refer to lighter, rather than to
darker (cf., ). And for depth, the deepest bowl was less wide than
the shallowest bowl, meaning that the identified 'more' pole (for
depth) corresponded to a 'less' pole (for width). To circumvent the
arbitrariness of what we considered to be the 'more' vs. 'less' pole of
a feature, the direction of the poles of each dimension were explicitly
taught to children.

" Relations of matching directions could be both positive, and both
negative. However, we again only focused on the two-negative-
relations case - to be consistent with what was done in Experiment 3
with sinking objects.

The content of the third fact differed from participant to participant.
It was the fact for which participants consistently picked the 'less'
options.

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Review of Psychology Frontier

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across trials of a fact. Nevertheless, the effect of condition
was still visible: There were more 'congruent' participants in
the weak-local-constraints condition (80% of adults, 30% of
children) than in the strong-local-constraints condition (70%
of adults, 15% of children). And this finding might mask the
true effect of condition, given that participants - to be
congruent - had to guess a negative correlation in the weak-
local-constraints condition (and a positive correlation in the
strong-local-constraints condition). Young children rarely

produce negative correlations spontaneously (e.g.

[13, 18,30]

).If

such directional bias could have been balanced out, the effect
of condition might have been even stronger.

Weak Local
Constraints i

Strong Local
_ Constraints

(0

o
o

CD -

.9 "5

to c
o 2

Q-
O

0.25 0.5 0.75

<&

4)

>
Q.

c

0.25 0.5 0.75

Relation 1 Trials:
Proportion of 'More' Choices

Fig. 10 Scatterplots of individual participants' responses in Experiment 4,

measured in proportion of 'more' choices for two of the three facts (referred

to as Relations 1 and 2), separated by condition and age group

Performance on the third fact is not displayed here. It was the fact that a

participant believed to be a negative relation. Given that the direction of the

third fact was fixed, we are able to denote the corners of the scatterplot that

correspond to congruent facts (marked with red circles).

In sum, the findings of this experiment provide further
support for our conceptualization of abduction. In a context in
which feature correlations were arbitrary, and participants had
to guess one of the facts - one that was not presented during
training - we found a stronger bias toward congruent facts in
the weak-local-constraints condition than the strong-local-
constraints condition for both children and adults. Children's
bias towards congruent facts was weaker overall (possibly
because they had difficulty with the arbitrariness of the facts),
while adults were biased by congruence in both conditions.

E. Experiment 5: Conceptual Replication II

In this final experiment, we return to a domain of plausible
facts to strengthen the manipulation of weak versus strong
local constraints. The goal was to replicate the findings of
Experiment 3 with a set of feature correlations for which
children are unlikely to have strong a-priori beliefs. Rather
than using sinking objects that differ in mass and volume, we
used composite objects that differed in two measures of
extension, each of which affected the size of the shadow cast
by the object. Facts about how the size of an object affects the

size of its shadow are physically meaningful, yet children are
unlikely to have strong beliefs about these facts.

The left side of Figure 11 shows schematics of the two
settings used in this experiment. In both settings, the outcome
feature pertains to the size of the shadow cast by a disc
(represented schematically with an arrow on the projection
screen). And in both settings, the size of the shadow was
affected by the sizes of two shapes, attached perpendicularly
to each other. In the setting shown in the top row of Figure 11,
the two shapes pertained to the projected disc and a 'base'
(which determined the distance between disc and light source.
And in the setting shown in the bottom row of Figure 1 1, the
two shapes pertained to the base (which again determined the
distance between disc and light source) and a 'tower' (which
determined the height of the light source). The arrows on
these shapes, shown in Figure 11, represent how their size
varied. The sizes of the two shapes affect shadow size either
in opposite ways (weak local constraints, top row of Figure
11), or in matching ways (strong local constraints; bottom row
of Figure 11).

The method was conceptually similar to that of
Experiment 3: A demonstration phase showed preschoolers
how the composite shapes affect the size of the projected
shadow. Children were shown a set of composite shapes for
which the sizes of the two component shapes were correlated.
In particular, the sizes correlated either positively (e.g., more
disc is more base) or negatively (e.g., more disc is less base).
The correlation was chosen to be either congruent or
incongruent with the two shadow facts. For example, in the
incongruent-facts and the weak-local-constraints condition
(top left quadrant of Figure 11), the three facts were: more
disc is more base, more disc is more shadow, and more base
is less shadow. And in the congruent-facts and the strong-
local-constraints condition (bottom right quadrant of Figure
11), the three facts were: more base is more tower, more base
is less shadow, and more tower is less shadow.

Incongruent Facts

Congruent Facts

Fig. 1 1 Schematic representation of the conditions in Experiment 5

Conditions differ in whether children are asked to learn two opposite facts

(yielding weak local constraints; top row) or two matching facts (yielding

strong local constraints; bottom row). And they differ in whether the facts are

congruent among each other (right column) or incongruent (left column).

Testing consisted of asking children to design a composite
object that would make a bigger shadow than a standard. As
was done in Experiment 3, children were given a choice for
only one of the shape sizes, while the other shape size was
held constant (analogous to mass trials and volume trials).

The manipulation resulted in a 2x2 design, with cells
differing in weak versus strong local constraints (shadow facts
had either opposite or matching directions, respectively), and

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Review of Psychology Frontier

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cells differing in congruent versus incongruent triads of facts.
In the incongruent case, when local facts are pitted against
global congruence, the prediction was that the strong-local-
constraints conditions yield A > 1, while the weak-local-
constraints conditions yield A < 1. In the congruent case,
when local facts agreed with global congruence, both
conditions yield A = 1, thus serving as control conditions to
establish whether children can learn the shadow facts
presented to them.

Figure 12 shows children's performance in scatter plots
corresponding to weak versus strong local constraints and
globally congruent versus incongruent facts. Red circles
indicate performance consistent with congruent facts, and
gray arrows indicate correct performance, consistent with
demonstrations. Note that red circles coincide with gray
arrows in the congruent-facts cases, but not in the
incongruent-facts cases. As can be seen in the figure, many
children performed correctly in strong-local constraints
settings (68%), but not in the weak-local-constraints settings
(19%). More importantly, and in line with our predictions,
performance in the incongruent-facts condition reflected a
bias toward globally congruent facts in the weak-local-
constraints condition (for 62% of children), but not in the
strong-local-constraints condition (0% of children).

Incongruent Facts Congruent Facts

® Q - ■

2?

9

0.25 0.5 0.75 1 g

Disc Trials

Proportion of More' Choices

0.25 0.5 0.75

-*

Weak Local
Constraints

Strong Local
Constraints

TowerTrials:
Proportion of More' Choices

Fig. 12 Scatterplots of individual children's responses in Exp. 5, measured in
proportion of more' choices on each trial type

The red circles denote the corners of the scatterplot that correspond to

congruent facts. And the grey arrows denote the corners of the scatterplot that

correspond to correct facts.

Taken together, this series of five experiments reveal that
a bias exists toward higher-order congruence that can be
captured by changes in a ratio of constraints (values of a
control parameter) toward local versus global order. Overall,
this ratio predicted and organized the performance of children
and adults, to cause-effect facts as well as to arbitrary feature
correlations in several task contexts.

III. CONCLUSION

Previous findings attest to a curious interplay between a
child's ability to abstract a higher-order pattern of Gestalt and
the detection of elements that compose those higher-order
patterns. Quinn et al. [25] for example, found that infants who
could attend to the higher-order pattern also remembered the
individual shapes better (Experiment 3). This Ying-Yang of

attention to local and global patterns - though far from what is
traditionally referred to as abduction - has an important
relation to abduction. It reflects the interdependence of
constraints on a local level with constraints on a more abstract
level. The current paper exploited this interplay in defining a
ratio of constraints that could capture the emergent outcomes
of abduction.

Changes in constraints have been shown to capture
systemic changes in systems of very different material
composition, ranging from physical models of fluids and
solids, to chemical substances, to the bodily interactions as an
infant is learning to walk (for a review, see [201 ). Here we
applied it to the most cognitive of human activities: abductive
reasoning. As such, we adapted an established method from
nonlinear dynamics to a system that has traditionally been
explained by mental constructs alone. Such mental constructs,
say the hierarchical representation of a knowledge domain, or
the mental process of analogy, are intuitive. But they fail to
capture what one might consider the essence of systemic order:
the nature of compromise among competing forces (cf. [231 ).
The idea of a control hierarchy and its entailed control
parameters fills this gap. It reveals abduction as the results of
competing sources of constraint, favoring one or another
hypothesis.

The empirical studies reviewed here illustrate how
children's abductive hypotheses about the physics of a
laboratory world can motivate a nonlinear dynamical
description of abduction. The description centers on
demonstrated local constraints that may increase or decrease
the degrees of freedom for choices - which in turn open the
possibility of observed choices reflecting the bias of induction
toward globally congruent facts (cf., [ 61 ). Though merely a
starting point, our review could provide a common umbrella
for notorious context dependence of abductive inferences. It
emphasizes the precarious balance of sources of knowledge
that determine whether induction will result in a reliable or
spurious higher-order Gestalt in thought and behavior. As
such, our proposal may show the promise to reconcile
conflicting findings about children's higher-order reasoning.

ACKNOWLEDGMENT

Findings reported in this paper and preparation of the
manuscript was funded in part by the grants from NICHD to
HK (HD055324) and from NSF to HK (DRL #723638) and
GVO (DHB #0728743; BCS #0642716). We are grateful to
Roger Schvaneveldt for comments and suggestions on earlier
version of this paper. And we thank Ramon D. Castillo for his
editorial help.

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